# Properties

 Label 1859.4.a.e Level $1859$ Weight $4$ Character orbit 1859.a Self dual yes Analytic conductor $109.685$ Analytic rank $0$ Dimension $11$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1859 = 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1859.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.684550701$$ Analytic rank: $$0$$ Dimension: $$11$$ Coefficient field: $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ Defining polynomial: $$x^{11} - 5 x^{10} - 64 x^{9} + 268 x^{8} + 1564 x^{7} - 4963 x^{6} - 16942 x^{5} + 37082 x^{4} + 68209 x^{3} - 90926 x^{2} - 1672 x + 16256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 143) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{10}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 1 - \beta_{5} ) q^{3} + ( 6 - \beta_{1} + \beta_{2} ) q^{4} + \beta_{8} q^{5} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{6} + ( -4 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{7} + ( -9 + 7 \beta_{1} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{8} + ( 14 - 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{2} + ( 1 - \beta_{5} ) q^{3} + ( 6 - \beta_{1} + \beta_{2} ) q^{4} + \beta_{8} q^{5} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{6} + ( -4 - \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{7} + ( -9 + 7 \beta_{1} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{10} ) q^{8} + ( 14 - 2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{9} + ( 7 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{10} -11 q^{11} + ( 10 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 6 \beta_{5} + 3 \beta_{6} + 3 \beta_{8} - 2 \beta_{10} ) q^{12} + ( -2 - 6 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} - \beta_{9} + \beta_{10} ) q^{14} + ( 9 + 6 \beta_{1} + 2 \beta_{2} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} ) q^{15} + ( 35 - 6 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{16} + ( 29 + 3 \beta_{1} - 3 \beta_{2} - \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + 4 \beta_{7} - 6 \beta_{8} - 4 \beta_{9} - \beta_{10} ) q^{17} + ( -37 + 21 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - 11 \beta_{5} + 3 \beta_{7} - 3 \beta_{8} - 4 \beta_{9} ) q^{18} + ( -4 - 12 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - 3 \beta_{10} ) q^{19} + ( 12 - 9 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{4} - 10 \beta_{5} + \beta_{6} + 2 \beta_{7} + 5 \beta_{8} - \beta_{9} - 4 \beta_{10} ) q^{20} + ( 29 - 19 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 5 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} ) q^{21} + ( 11 - 11 \beta_{1} ) q^{22} + ( 1 - 7 \beta_{1} - 3 \beta_{2} + 4 \beta_{4} + 9 \beta_{5} - \beta_{6} - 3 \beta_{7} + 3 \beta_{8} + \beta_{9} - 4 \beta_{10} ) q^{23} + ( 10 + 10 \beta_{1} - 11 \beta_{3} + 8 \beta_{5} - 10 \beta_{6} + 7 \beta_{7} - 6 \beta_{8} - 6 \beta_{9} + 5 \beta_{10} ) q^{24} + ( 59 + 28 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} - 3 \beta_{8} - 8 \beta_{9} ) q^{25} + ( 3 + 32 \beta_{1} - 8 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 17 \beta_{5} + 6 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{10} ) q^{27} + ( -36 - 27 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} - 5 \beta_{4} - 29 \beta_{5} + 8 \beta_{6} - 3 \beta_{7} + 8 \beta_{8} - 2 \beta_{9} - 5 \beta_{10} ) q^{28} + ( 35 - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} - 7 \beta_{10} ) q^{29} + ( 58 + 51 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} + \beta_{4} - 7 \beta_{5} - 7 \beta_{6} + 15 \beta_{7} - 8 \beta_{8} - 6 \beta_{9} + 7 \beta_{10} ) q^{30} + ( 8 - 13 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 8 \beta_{5} + 11 \beta_{6} - 8 \beta_{7} + 12 \beta_{8} + 9 \beta_{9} ) q^{31} + ( -39 + 16 \beta_{1} + 5 \beta_{2} - \beta_{3} + 9 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} - 7 \beta_{8} - \beta_{9} + 3 \beta_{10} ) q^{32} + ( -11 + 11 \beta_{5} ) q^{33} + ( -61 + 25 \beta_{1} + 11 \beta_{2} + 3 \beta_{3} + 11 \beta_{4} - 13 \beta_{5} - 5 \beta_{6} + 6 \beta_{7} + 6 \beta_{8} + 12 \beta_{9} + 5 \beta_{10} ) q^{34} + ( 60 - \beta_{1} + 13 \beta_{2} + 3 \beta_{3} - \beta_{4} + 6 \beta_{5} + 7 \beta_{6} - 18 \beta_{7} + 11 \beta_{8} + 7 \beta_{9} - 2 \beta_{10} ) q^{35} + ( 149 - 44 \beta_{1} + 17 \beta_{2} - 8 \beta_{3} - 7 \beta_{4} + 16 \beta_{5} - 8 \beta_{6} + 9 \beta_{7} + 7 \beta_{8} - 7 \beta_{9} - 4 \beta_{10} ) q^{36} + ( -5 + 12 \beta_{1} - 12 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} - \beta_{5} - 2 \beta_{6} + 8 \beta_{7} - 14 \beta_{8} - 13 \beta_{9} - \beta_{10} ) q^{37} + ( -171 - 7 \beta_{1} - 4 \beta_{2} + \beta_{3} + 13 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} + 13 \beta_{8} + 13 \beta_{9} + 5 \beta_{10} ) q^{38} + ( -112 + 29 \beta_{1} - 19 \beta_{2} - 11 \beta_{3} + 9 \beta_{4} - 22 \beta_{5} - 15 \beta_{6} + 18 \beta_{7} - 19 \beta_{8} - 17 \beta_{9} - 2 \beta_{10} ) q^{40} + ( -31 - 4 \beta_{1} + 16 \beta_{2} - 3 \beta_{3} + 9 \beta_{4} + 17 \beta_{5} - 7 \beta_{7} + 13 \beta_{8} + 12 \beta_{9} + 4 \beta_{10} ) q^{41} + ( -268 + 93 \beta_{1} - 9 \beta_{2} + 8 \beta_{3} + 15 \beta_{4} - 15 \beta_{5} - 2 \beta_{7} - 10 \beta_{8} + 9 \beta_{9} + 11 \beta_{10} ) q^{42} + ( 46 + 27 \beta_{1} + 15 \beta_{2} - \beta_{3} - 3 \beta_{4} - 12 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} + 11 \beta_{8} + 5 \beta_{9} + 4 \beta_{10} ) q^{43} + ( -66 + 11 \beta_{1} - 11 \beta_{2} ) q^{44} + ( 145 + 16 \beta_{1} + 18 \beta_{2} - 6 \beta_{3} - 15 \beta_{5} - 4 \beta_{6} + 16 \beta_{7} - 19 \beta_{8} - 17 \beta_{9} - 3 \beta_{10} ) q^{45} + ( -40 - 19 \beta_{1} - 31 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} - 13 \beta_{5} - \beta_{6} - 7 \beta_{7} - 4 \beta_{8} - 11 \beta_{9} - 10 \beta_{10} ) q^{46} + ( 49 - 21 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} - 7 \beta_{4} - \beta_{5} - 7 \beta_{6} - 12 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} - \beta_{10} ) q^{47} + ( -13 - 44 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} - 8 \beta_{4} - 60 \beta_{5} + 21 \beta_{6} - 13 \beta_{7} + 20 \beta_{8} + \beta_{9} - 9 \beta_{10} ) q^{48} + ( 131 - 27 \beta_{1} + 13 \beta_{2} + 11 \beta_{3} - 9 \beta_{4} + 40 \beta_{5} + 11 \beta_{6} - 8 \beta_{7} + 25 \beta_{8} + 12 \beta_{9} + \beta_{10} ) q^{49} + ( 238 + 76 \beta_{1} + 42 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} + \beta_{7} + 35 \beta_{8} + 21 \beta_{9} + 8 \beta_{10} ) q^{50} + ( 3 + 43 \beta_{1} - 5 \beta_{2} - \beta_{3} - 9 \beta_{4} - 47 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} + 4 \beta_{8} - 10 \beta_{9} - 3 \beta_{10} ) q^{51} + ( -52 - 20 \beta_{2} - 4 \beta_{3} - 12 \beta_{4} + 4 \beta_{5} + 8 \beta_{6} + 4 \beta_{7} - 8 \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{53} + ( 349 - 49 \beta_{1} + 58 \beta_{2} - 17 \beta_{3} + 3 \beta_{4} + 69 \beta_{5} - 20 \beta_{6} + 11 \beta_{7} + 8 \beta_{8} + 3 \beta_{9} + 11 \beta_{10} ) q^{54} -11 \beta_{8} q^{55} + ( -253 - 26 \beta_{1} - 13 \beta_{2} - 33 \beta_{3} + 16 \beta_{4} + 46 \beta_{5} - 43 \beta_{6} + 25 \beta_{7} - 34 \beta_{8} - 18 \beta_{9} + 7 \beta_{10} ) q^{56} + ( -6 - 20 \beta_{1} - 10 \beta_{2} + 21 \beta_{3} - 15 \beta_{4} - 8 \beta_{5} + 18 \beta_{6} - 7 \beta_{7} + 10 \beta_{8} + 7 \beta_{9} - \beta_{10} ) q^{57} + ( 13 + 78 \beta_{1} - 2 \beta_{2} + 14 \beta_{4} - 15 \beta_{5} - 12 \beta_{6} + 6 \beta_{7} - 27 \beta_{8} - 7 \beta_{9} + 5 \beta_{10} ) q^{58} + ( 27 + 33 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} - 7 \beta_{4} + 37 \beta_{5} - 11 \beta_{6} - 10 \beta_{7} - 3 \beta_{8} - 5 \beta_{10} ) q^{59} + ( 445 + 91 \beta_{1} + 21 \beta_{2} + 9 \beta_{3} - 13 \beta_{4} - 93 \beta_{5} + 17 \beta_{6} + 18 \beta_{7} + 8 \beta_{8} - 10 \beta_{9} + \beta_{10} ) q^{60} + ( 137 - 32 \beta_{1} + 28 \beta_{2} - 6 \beta_{3} + 16 \beta_{4} + 23 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} + 3 \beta_{8} + 5 \beta_{9} - 3 \beta_{10} ) q^{61} + ( -131 + 34 \beta_{1} + 23 \beta_{2} + \beta_{3} + 4 \beta_{4} + 76 \beta_{5} - 4 \beta_{6} - 9 \beta_{7} - 6 \beta_{8} + 6 \beta_{9} + 22 \beta_{10} ) q^{62} + ( -139 + 32 \beta_{1} - 20 \beta_{2} + 37 \beta_{3} + 5 \beta_{4} - 39 \beta_{5} + 36 \beta_{6} - 7 \beta_{7} + 9 \beta_{8} + 15 \beta_{9} - 11 \beta_{10} ) q^{63} + ( -86 - 37 \beta_{1} - \beta_{2} + 9 \beta_{3} - 18 \beta_{4} - 6 \beta_{5} + 14 \beta_{6} - 31 \beta_{7} + 22 \beta_{8} + 12 \beta_{9} - 16 \beta_{10} ) q^{64} + ( -11 - 22 \beta_{1} + 11 \beta_{3} - 11 \beta_{5} + 11 \beta_{6} - 11 \beta_{7} + 11 \beta_{8} + 11 \beta_{9} ) q^{66} + ( -9 - 33 \beta_{1} + 7 \beta_{2} - 21 \beta_{3} + 3 \beta_{4} - 41 \beta_{5} - 11 \beta_{6} - 6 \beta_{7} - 11 \beta_{8} - 24 \beta_{9} - 5 \beta_{10} ) q^{67} + ( 291 - \beta_{1} + 11 \beta_{2} - 15 \beta_{3} - 31 \beta_{4} - 17 \beta_{5} + 5 \beta_{6} + 12 \beta_{7} - 28 \beta_{8} - 44 \beta_{9} - 11 \beta_{10} ) q^{68} + ( -236 - 3 \beta_{1} + 37 \beta_{2} + 18 \beta_{3} + 18 \beta_{4} + 28 \beta_{5} - 13 \beta_{6} - 13 \beta_{7} - 4 \beta_{8} + 7 \beta_{9} - 2 \beta_{10} ) q^{69} + ( 52 + 135 \beta_{1} + 45 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} + 114 \beta_{5} - 11 \beta_{6} - 26 \beta_{7} + 5 \beta_{8} + 25 \beta_{9} + 24 \beta_{10} ) q^{70} + ( 143 + 18 \beta_{1} + 28 \beta_{2} - 6 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 14 \beta_{7} + 8 \beta_{8} + 29 \beta_{9} + \beta_{10} ) q^{71} + ( -299 + 221 \beta_{1} - 65 \beta_{2} - 7 \beta_{3} + 26 \beta_{4} - 114 \beta_{5} - \beta_{6} + 4 \beta_{7} - 10 \beta_{8} - 15 \beta_{9} + 7 \beta_{10} ) q^{72} + ( -13 - 2 \beta_{1} - 2 \beta_{2} - 22 \beta_{3} + 8 \beta_{4} - 17 \beta_{5} - 28 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + 16 \beta_{10} ) q^{73} + ( -22 - 93 \beta_{1} + 12 \beta_{2} + 14 \beta_{3} + 19 \beta_{4} - 45 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} + 52 \beta_{8} + 34 \beta_{9} - 5 \beta_{10} ) q^{74} + ( 156 + 84 \beta_{1} - 28 \beta_{3} - 2 \beta_{4} - 16 \beta_{5} - 10 \beta_{6} + 50 \beta_{7} - 20 \beta_{8} - 37 \beta_{9} - 13 \beta_{10} ) q^{75} + ( 281 - 186 \beta_{1} - 55 \beta_{2} + 11 \beta_{3} - 38 \beta_{4} - 13 \beta_{5} + 18 \beta_{6} - 5 \beta_{8} - 34 \beta_{9} - 21 \beta_{10} ) q^{76} + ( 44 + 11 \beta_{1} + 11 \beta_{2} - 11 \beta_{6} + 11 \beta_{7} - 11 \beta_{8} + 11 \beta_{10} ) q^{77} + ( 160 - 72 \beta_{1} - 6 \beta_{2} + 16 \beta_{3} - 12 \beta_{4} + 14 \beta_{5} + 2 \beta_{6} - 16 \beta_{7} + 18 \beta_{8} + 16 \beta_{9} - 6 \beta_{10} ) q^{79} + ( 274 - 167 \beta_{1} - 33 \beta_{2} + 3 \beta_{3} - 11 \beta_{4} - 70 \beta_{5} - 3 \beta_{6} + 20 \beta_{7} - 19 \beta_{8} - 33 \beta_{9} - 20 \beta_{10} ) q^{80} + ( 279 - 70 \beta_{1} + 32 \beta_{2} - 39 \beta_{3} - 17 \beta_{4} + 44 \beta_{5} - 38 \beta_{6} + 17 \beta_{7} - 30 \beta_{8} - 26 \beta_{9} + 20 \beta_{10} ) q^{81} + ( 111 + 35 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} - 43 \beta_{4} - 5 \beta_{5} + 43 \beta_{6} - 8 \beta_{7} - 4 \beta_{8} - 25 \beta_{9} - 8 \beta_{10} ) q^{82} + ( -108 + 32 \beta_{1} - 20 \beta_{2} + 13 \beta_{3} - 11 \beta_{4} + 20 \beta_{5} + 4 \beta_{6} + \beta_{7} + 8 \beta_{8} + 11 \beta_{9} - 23 \beta_{10} ) q^{83} + ( 1106 - 341 \beta_{1} + 69 \beta_{2} - \beta_{3} - 69 \beta_{4} + 111 \beta_{5} + 44 \beta_{6} - 25 \beta_{7} + 28 \beta_{8} + 29 \beta_{9} - 33 \beta_{10} ) q^{84} + ( -289 + 9 \beta_{1} + 31 \beta_{2} + 13 \beta_{3} - 11 \beta_{4} - 77 \beta_{5} - 3 \beta_{6} + 54 \beta_{7} + 16 \beta_{8} + 26 \beta_{9} + 9 \beta_{10} ) q^{85} + ( 496 + 169 \beta_{1} + 17 \beta_{2} - 13 \beta_{3} + 15 \beta_{4} - 14 \beta_{5} - 7 \beta_{6} + 10 \beta_{7} - 37 \beta_{8} - 39 \beta_{9} + 2 \beta_{10} ) q^{86} + ( 42 + 20 \beta_{1} - 6 \beta_{2} + 24 \beta_{3} - 10 \beta_{4} - 106 \beta_{5} - 8 \beta_{6} - 6 \beta_{7} + 4 \beta_{8} - 12 \beta_{9} + 4 \beta_{10} ) q^{87} + ( 99 - 77 \beta_{1} + 11 \beta_{5} - 11 \beta_{7} + 11 \beta_{8} - 11 \beta_{10} ) q^{88} + ( -152 - 67 \beta_{1} + 27 \beta_{2} + 5 \beta_{3} + 25 \beta_{4} + 26 \beta_{5} - 7 \beta_{6} - 38 \beta_{7} + 14 \beta_{8} + 13 \beta_{9} - 30 \beta_{10} ) q^{89} + ( -79 + 376 \beta_{1} + 26 \beta_{2} - 10 \beta_{3} + 8 \beta_{4} - 119 \beta_{5} - 10 \beta_{6} + 64 \beta_{7} - 5 \beta_{8} + 5 \beta_{9} + 21 \beta_{10} ) q^{90} + ( -217 - 202 \beta_{1} - 27 \beta_{2} - 19 \beta_{3} + 16 \beta_{4} - 40 \beta_{5} - 37 \beta_{6} - 5 \beta_{7} + 16 \beta_{8} + 2 \beta_{9} + 10 \beta_{10} ) q^{92} + ( -40 - 251 \beta_{1} + 5 \beta_{2} + 19 \beta_{3} - 33 \beta_{4} - 50 \beta_{5} + 7 \beta_{6} - 8 \beta_{7} + 22 \beta_{8} + 15 \beta_{9} + 26 \beta_{10} ) q^{93} + ( -253 - 65 \beta_{1} - 25 \beta_{2} + 21 \beta_{3} + 39 \beta_{4} + 33 \beta_{5} - 3 \beta_{6} - 56 \beta_{7} + 44 \beta_{8} + 34 \beta_{9} - 15 \beta_{10} ) q^{94} + ( -302 - 70 \beta_{1} + 66 \beta_{2} + 24 \beta_{3} - 10 \beta_{4} + 44 \beta_{5} + 32 \beta_{6} - 60 \beta_{7} + 68 \beta_{8} + 82 \beta_{9} - 14 \beta_{10} ) q^{95} + ( -474 + 28 \beta_{1} + 12 \beta_{2} - 20 \beta_{3} + 48 \beta_{4} + 152 \beta_{5} - 58 \beta_{6} + 10 \beta_{7} - 26 \beta_{8} + 15 \beta_{10} ) q^{96} + ( -12 - 3 \beta_{1} - 7 \beta_{2} + 9 \beta_{3} - \beta_{4} - 134 \beta_{5} - 11 \beta_{6} + 40 \beta_{7} - 40 \beta_{8} - 37 \beta_{9} + 12 \beta_{10} ) q^{97} + ( -374 + 263 \beta_{1} + 7 \beta_{2} + 10 \beta_{3} + 36 \beta_{4} + 36 \beta_{5} - \beta_{6} - 26 \beta_{7} - 26 \beta_{8} + 10 \beta_{9} + 43 \beta_{10} ) q^{98} + ( -154 + 22 \beta_{1} + 11 \beta_{3} + 11 \beta_{4} + 11 \beta_{5} - 11 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$11 q - 6 q^{2} + 6 q^{3} + 66 q^{4} + 4 q^{5} + 14 q^{6} - 45 q^{7} - 78 q^{8} + 135 q^{9} + O(q^{10})$$ $$11 q - 6 q^{2} + 6 q^{3} + 66 q^{4} + 4 q^{5} + 14 q^{6} - 45 q^{7} - 78 q^{8} + 135 q^{9} + 48 q^{10} - 121 q^{11} + 105 q^{12} - 48 q^{14} + 125 q^{15} + 394 q^{16} + 265 q^{17} - 405 q^{18} - 127 q^{19} + 46 q^{20} + 287 q^{21} + 66 q^{22} + 42 q^{23} + 83 q^{24} + 737 q^{25} + 69 q^{27} - 675 q^{28} + 435 q^{29} + 785 q^{30} + 174 q^{31} - 315 q^{32} - 66 q^{33} - 497 q^{34} + 844 q^{35} + 1572 q^{36} - 187 q^{37} - 1813 q^{38} - 1470 q^{40} - 128 q^{41} - 2630 q^{42} + 696 q^{43} - 726 q^{44} + 1537 q^{45} - 785 q^{46} + 355 q^{47} - 516 q^{48} + 1758 q^{49} + 3414 q^{50} - 25 q^{51} - 693 q^{53} + 4150 q^{54} - 44 q^{55} - 3123 q^{56} - 99 q^{57} + 287 q^{58} + 609 q^{59} + 5013 q^{60} + 1625 q^{61} - 882 q^{62} - 1365 q^{63} - 914 q^{64} - 154 q^{66} - 633 q^{67} + 2873 q^{68} - 2192 q^{69} + 2054 q^{70} + 1937 q^{71} - 3242 q^{72} - 404 q^{73} - 447 q^{74} + 1781 q^{75} + 1814 q^{76} + 495 q^{77} + 1670 q^{79} + 1568 q^{80} + 2619 q^{81} + 1283 q^{82} - 785 q^{83} + 11750 q^{84} - 3189 q^{85} + 5950 q^{86} + 46 q^{87} + 858 q^{88} - 1464 q^{89} + 401 q^{90} - 3786 q^{92} - 1826 q^{93} - 2597 q^{94} - 2356 q^{95} - 4513 q^{96} - 1184 q^{97} - 2823 q^{98} - 1485 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{11} - 5 x^{10} - 64 x^{9} + 268 x^{8} + 1564 x^{7} - 4963 x^{6} - 16942 x^{5} + 37082 x^{4} + 68209 x^{3} - 90926 x^{2} - 1672 x + 16256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 13$$ $$\beta_{3}$$ $$=$$ $$($$$$-73921 \nu^{10} + 8439311 \nu^{9} + 2434932 \nu^{8} - 603336064 \nu^{7} - 474655656 \nu^{6} + 14205752999 \nu^{5} + 19340214652 \nu^{4} - 109768364140 \nu^{3} - 177507724291 \nu^{2} + 115473527850 \nu + 47192465320$$$$)/ 601708456$$ $$\beta_{4}$$ $$=$$ $$($$$$130026 \nu^{10} - 1931719 \nu^{9} - 3195319 \nu^{8} + 103863653 \nu^{7} + 20780639 \nu^{6} - 1917770081 \nu^{5} - 841766832 \nu^{4} + 11928784725 \nu^{3} + 11241059270 \nu^{2} - 5167754840 \nu - 4999000508$$$$)/ 150427114$$ $$\beta_{5}$$ $$=$$ $$($$$$-833811 \nu^{10} + 2410905 \nu^{9} + 50444340 \nu^{8} - 75562068 \nu^{7} - 1065164652 \nu^{6} + 240341729 \nu^{5} + 7571043660 \nu^{4} + 3804215076 \nu^{3} - 5610904205 \nu^{2} + 6664162042 \nu - 1290598552$$$$)/ 601708456$$ $$\beta_{6}$$ $$=$$ $$($$$$-499260 \nu^{10} - 759762 \nu^{9} + 34818189 \nu^{8} + 92778516 \nu^{7} - 721641868 \nu^{6} - 2879270108 \nu^{5} + 2636047738 \nu^{4} + 24702995781 \nu^{3} + 25591814581 \nu^{2} - 17867141655 \nu - 8760781816$$$$)/ 150427114$$ $$\beta_{7}$$ $$=$$ $$($$$$2635821 \nu^{10} - 5571119 \nu^{9} - 174619828 \nu^{8} + 157660112 \nu^{7} + 3998941440 \nu^{6} + 458087445 \nu^{5} - 32353972432 \nu^{4} - 21467545304 \nu^{3} + 54801353963 \nu^{2} + 12748743462 \nu - 11305102248$$$$)/ 601708456$$ $$\beta_{8}$$ $$=$$ $$($$$$-3965935 \nu^{10} + 6054403 \nu^{9} + 281696508 \nu^{8} - 144928752 \nu^{7} - 6964105264 \nu^{6} - 1548015215 \nu^{5} + 64316625494 \nu^{4} + 28341677850 \nu^{3} - 168377556283 \nu^{2} + 35533460802 \nu + 35528598512$$$$)/ 601708456$$ $$\beta_{9}$$ $$=$$ $$($$$$6914567 \nu^{10} - 19945349 \nu^{9} - 450124744 \nu^{8} + 773726616 \nu^{7} + 10426407568 \nu^{6} - 7237952209 \nu^{5} - 91831408952 \nu^{4} + 12409668156 \nu^{3} + 229168439269 \nu^{2} - 68872700658 \nu - 44225190288$$$$)/ 601708456$$ $$\beta_{10}$$ $$=$$ $$($$$$-7435567 \nu^{10} + 14036427 \nu^{9} + 506760676 \nu^{8} - 378150932 \nu^{7} - 12028211356 \nu^{6} - 1765760931 \nu^{5} + 104241641586 \nu^{4} + 54215146686 \nu^{3} - 230594939819 \nu^{2} + 17414710262 \nu + 59984105152$$$$)/ 601708456$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 13$$ $$\nu^{3}$$ $$=$$ $$\beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + 3 \beta_{2} + 23 \beta_{1} + 15$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 32 \beta_{2} + 60 \beta_{1} + 288$$ $$\nu^{5}$$ $$=$$ $$45 \beta_{10} + 9 \beta_{9} - 39 \beta_{8} + 45 \beta_{7} + 9 \beta_{6} - 25 \beta_{5} - 6 \beta_{4} + 9 \beta_{3} + 145 \beta_{2} + 635 \beta_{1} + 774$$ $$\nu^{6}$$ $$=$$ $$214 \beta_{10} + 116 \beta_{9} - 122 \beta_{8} + 224 \beta_{7} + 143 \beta_{6} - 91 \beta_{5} - 129 \beta_{4} + 113 \beta_{3} + 1090 \beta_{2} + 2508 \beta_{1} + 7758$$ $$\nu^{7}$$ $$=$$ $$1716 \beta_{10} + 657 \beta_{9} - 1316 \beta_{8} + 1541 \beta_{7} + 577 \beta_{6} - 422 \beta_{5} - 422 \beta_{4} + 602 \beta_{3} + 5824 \beta_{2} + 19607 \beta_{1} + 31051$$ $$\nu^{8}$$ $$=$$ $$9061 \beta_{10} + 5572 \beta_{9} - 5389 \beta_{8} + 7817 \beta_{7} + 5650 \beta_{6} - 1185 \beta_{5} - 4716 \beta_{4} + 5180 \beta_{3} + 38628 \beta_{2} + 94891 \beta_{1} + 237000$$ $$\nu^{9}$$ $$=$$ $$63313 \beta_{10} + 34338 \beta_{9} - 44023 \beta_{8} + 48504 \beta_{7} + 27533 \beta_{6} + 1418 \beta_{5} - 20585 \beta_{4} + 30186 \beta_{3} + 221398 \beta_{2} + 645848 \beta_{1} + 1155998$$ $$\nu^{10}$$ $$=$$ $$356210 \beta_{10} + 249413 \beta_{9} - 212170 \beta_{8} + 250295 \beta_{7} + 216294 \beta_{6} + 47171 \beta_{5} - 170765 \beta_{4} + 222509 \beta_{3} + 1396186 \beta_{2} + 3461508 \beta_{1} + 7773705$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −4.35762 −4.05967 −3.37601 −3.16556 −0.390260 0.636678 0.778412 3.58491 3.59635 5.65354 6.09923
−5.35762 7.47772 20.7041 10.6979 −40.0628 32.3181 −68.0635 28.9163 −57.3151
1.2 −5.05967 −10.2152 17.6003 4.34036 51.6854 −31.2716 −48.5744 77.3494 −21.9608
1.3 −4.37601 4.08546 11.1495 −1.30050 −17.8780 −14.9266 −13.7821 −10.3090 5.69101
1.4 −4.16556 −0.469632 9.35186 −20.3915 1.95628 −2.83667 −5.63125 −26.7794 84.9419
1.5 −1.39026 8.98097 −6.06718 −6.95692 −12.4859 −9.63375 19.5570 53.6578 9.67193
1.6 −0.363322 −3.74669 −7.86800 11.8947 1.36126 −28.3874 5.76520 −12.9623 −4.32160
1.7 −0.221588 −7.13795 −7.95090 −6.37060 1.58168 23.0480 3.53453 23.9503 1.41165
1.8 2.58491 −2.54183 −1.31822 20.1048 −6.57040 29.2165 −24.0868 −20.5391 51.9693
1.9 2.59635 4.20666 −1.25897 −11.3514 10.9220 −3.36670 −24.0395 −9.30403 −29.4723
1.10 4.65354 8.62383 13.6554 21.5680 40.1313 −9.18213 26.3178 47.3704 100.367
1.11 5.09923 −3.26338 18.0021 −18.2348 −16.6407 −29.9777 51.0031 −16.3503 −92.9834
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$1$$
$$13$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1859.4.a.e 11
13.b even 2 1 143.4.a.d 11
39.d odd 2 1 1287.4.a.m 11
52.b odd 2 1 2288.4.a.u 11
143.d odd 2 1 1573.4.a.f 11

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.a.d 11 13.b even 2 1
1287.4.a.m 11 39.d odd 2 1
1573.4.a.f 11 143.d odd 2 1
1859.4.a.e 11 1.a even 1 1 trivial
2288.4.a.u 11 52.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{11} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1859))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$8808 + 67420 T + 130202 T^{2} + 11794 T^{3} - 57357 T^{4} - 7730 T^{5} + 7525 T^{6} + 1134 T^{7} - 368 T^{8} - 59 T^{9} + 6 T^{10} + T^{11}$$
$3$ $$-10592776 - 26381750 T - 6141153 T^{2} + 4884358 T^{3} + 1149514 T^{4} - 360777 T^{5} - 62128 T^{6} + 12882 T^{7} + 1129 T^{8} - 198 T^{9} - 6 T^{10} + T^{11}$$
$5$ $$-58263405696 - 44138856448 T + 4350144704 T^{2} + 3027663456 T^{3} - 11853488 T^{4} - 54061188 T^{5} - 428566 T^{6} + 375867 T^{7} + 2698 T^{8} - 1048 T^{9} - 4 T^{10} + T^{11}$$
$7$ $$-7302898028448 - 6727883017652 T - 2197122360352 T^{2} - 297699209589 T^{3} - 12214946131 T^{4} + 785132181 T^{5} + 73682462 T^{6} + 267354 T^{7} - 104422 T^{8} - 1753 T^{9} + 45 T^{10} + T^{11}$$
$11$ $$( 11 + T )^{11}$$
$13$ $$T^{11}$$
$17$ $$-24188667748937367552 + 4984502135268605952 T - 423844914020930560 T^{2} + 19074784887786240 T^{3} - 476600090667968 T^{4} + 5911837779360 T^{5} - 8099837696 T^{6} - 678834176 T^{7} + 6707568 T^{8} - 1386 T^{9} - 265 T^{10} + T^{11}$$
$19$ $$217809390543486976 - 492105377617548288 T + 47578053402485808 T^{2} + 829222431137620 T^{3} - 128985209540412 T^{4} - 701918978235 T^{5} + 50445861335 T^{6} + 367350546 T^{7} - 4679841 T^{8} - 36510 T^{9} + 127 T^{10} + T^{11}$$
$23$ $$72247540401681857184 - 12853011646171792508 T - 557508639020137449 T^{2} + 19729600771945246 T^{3} + 467943626648130 T^{4} - 8419511970909 T^{5} - 81042715212 T^{6} + 1276222260 T^{7} + 3559669 T^{8} - 63586 T^{9} - 42 T^{10} + T^{11}$$
$29$ $$-$$$$37\!\cdots\!28$$$$+$$$$23\!\cdots\!08$$$$T - 13185943386695049728 T^{2} - 901915922935533184 T^{3} + 8016671003564416 T^{4} + 100102138632928 T^{5} - 1073425721856 T^{6} - 2461072880 T^{7} + 45548876 T^{8} - 58094 T^{9} - 435 T^{10} + T^{11}$$
$31$ $$39\!\cdots\!84$$$$-$$$$43\!\cdots\!44$$$$T -$$$$73\!\cdots\!64$$$$T^{2} + 7142524863831497088 T^{3} + 80648215149129888 T^{4} - 500564381377456 T^{5} - 2950247951008 T^{6} + 16088152228 T^{7} + 40946072 T^{8} - 220267 T^{9} - 174 T^{10} + T^{11}$$
$37$ $$21\!\cdots\!32$$$$-$$$$81\!\cdots\!24$$$$T - 29460962275876355072 T^{2} + 1118845908828217856 T^{3} - 3607903109944576 T^{4} - 140943855612688 T^{5} + 721662789700 T^{6} + 6980478356 T^{7} - 27539835 T^{8} - 167041 T^{9} + 187 T^{10} + T^{11}$$
$41$ $$61\!\cdots\!72$$$$-$$$$19\!\cdots\!04$$$$T -$$$$31\!\cdots\!36$$$$T^{2} + 56114279680678427262 T^{3} - 32279892823616551 T^{4} - 2682257535293520 T^{5} + 3727475384449 T^{6} + 48888881297 T^{7} - 46542835 T^{8} - 385757 T^{9} + 128 T^{10} + T^{11}$$
$43$ $$-$$$$91\!\cdots\!44$$$$-$$$$14\!\cdots\!56$$$$T +$$$$53\!\cdots\!64$$$$T^{2} + 14814633639727330816 T^{3} - 1162790805969910656 T^{4} + 9928595838556880 T^{5} - 26632551187384 T^{6} - 38181942108 T^{7} + 297660134 T^{8} - 252207 T^{9} - 696 T^{10} + T^{11}$$
$47$ $$11\!\cdots\!76$$$$+$$$$47\!\cdots\!40$$$$T -$$$$47\!\cdots\!00$$$$T^{2} +$$$$17\!\cdots\!44$$$$T^{3} + 2064792619584021440 T^{4} - 6989868281402912 T^{5} - 30208260137456 T^{6} + 94144527288 T^{7} + 176268664 T^{8} - 518364 T^{9} - 355 T^{10} + T^{11}$$
$53$ $$-$$$$58\!\cdots\!96$$$$-$$$$30\!\cdots\!48$$$$T -$$$$47\!\cdots\!24$$$$T^{2} -$$$$17\!\cdots\!16$$$$T^{3} + 1973382108291668026 T^{4} + 20916639340235641 T^{5} + 60397134637055 T^{6} - 32743085326 T^{7} - 432186529 T^{8} - 418854 T^{9} + 693 T^{10} + T^{11}$$
$59$ $$-$$$$13\!\cdots\!24$$$$-$$$$82\!\cdots\!72$$$$T +$$$$19\!\cdots\!32$$$$T^{2} +$$$$21\!\cdots\!12$$$$T^{3} + 180467774718702160 T^{4} - 64350828788025048 T^{5} - 76537637566756 T^{6} + 403385082740 T^{7} + 432787765 T^{8} - 1026973 T^{9} - 609 T^{10} + T^{11}$$
$61$ $$-$$$$15\!\cdots\!16$$$$+$$$$68\!\cdots\!56$$$$T -$$$$45\!\cdots\!16$$$$T^{2} -$$$$28\!\cdots\!36$$$$T^{3} + 19002461465518407296 T^{4} + 45963313493667008 T^{5} - 211627620331088 T^{6} - 207543489712 T^{7} + 980911596 T^{8} + 10688 T^{9} - 1625 T^{10} + T^{11}$$
$67$ $$30\!\cdots\!12$$$$+$$$$22\!\cdots\!68$$$$T -$$$$10\!\cdots\!48$$$$T^{2} -$$$$15\!\cdots\!68$$$$T^{3} + 35836452939054727152 T^{4} - 163072407740677656 T^{5} + 53562173278308 T^{6} + 809107737882 T^{7} - 559506101 T^{8} - 1476795 T^{9} + 633 T^{10} + T^{11}$$
$71$ $$34\!\cdots\!28$$$$-$$$$16\!\cdots\!32$$$$T -$$$$15\!\cdots\!44$$$$T^{2} +$$$$33\!\cdots\!12$$$$T^{3} -$$$$23\!\cdots\!56$$$$T^{4} + 673982379460970896 T^{5} - 509386410054688 T^{6} - 1340435168474 T^{7} + 2552641957 T^{8} - 246267 T^{9} - 1937 T^{10} + T^{11}$$
$73$ $$-$$$$21\!\cdots\!04$$$$-$$$$83\!\cdots\!36$$$$T -$$$$66\!\cdots\!56$$$$T^{2} +$$$$31\!\cdots\!82$$$$T^{3} - 4496506145894409629 T^{4} - 324839755326858096 T^{5} + 135565372456567 T^{6} + 1263299284441 T^{7} - 500295657 T^{8} - 1983653 T^{9} + 404 T^{10} + T^{11}$$
$79$ $$58\!\cdots\!08$$$$+$$$$90\!\cdots\!60$$$$T -$$$$41\!\cdots\!92$$$$T^{2} -$$$$86\!\cdots\!80$$$$T^{3} + 11332650006935288448 T^{4} + 201914510546754240 T^{5} - 424594497792480 T^{6} - 630175421536 T^{7} + 1994686400 T^{8} - 478268 T^{9} - 1670 T^{10} + T^{11}$$
$83$ $$-$$$$53\!\cdots\!32$$$$-$$$$23\!\cdots\!12$$$$T +$$$$20\!\cdots\!92$$$$T^{2} +$$$$64\!\cdots\!64$$$$T^{3} -$$$$23\!\cdots\!28$$$$T^{4} - 532104108037010357 T^{5} + 1104913439056357 T^{6} + 1821313226608 T^{7} - 1927236611 T^{8} - 2661500 T^{9} + 785 T^{10} + T^{11}$$
$89$ $$10\!\cdots\!48$$$$+$$$$12\!\cdots\!80$$$$T +$$$$20\!\cdots\!64$$$$T^{2} -$$$$16\!\cdots\!04$$$$T^{3} -$$$$52\!\cdots\!32$$$$T^{4} + 431182988724623200 T^{5} + 2584454143729760 T^{6} + 602758010236 T^{7} - 4004062738 T^{8} - 2311383 T^{9} + 1464 T^{10} + T^{11}$$
$97$ $$23\!\cdots\!28$$$$-$$$$33\!\cdots\!48$$$$T -$$$$28\!\cdots\!32$$$$T^{2} +$$$$26\!\cdots\!16$$$$T^{3} -$$$$13\!\cdots\!68$$$$T^{4} - 8503242277053197696 T^{5} + 5344682865992184 T^{6} + 9834390006056 T^{7} - 4609737758 T^{8} - 5081683 T^{9} + 1184 T^{10} + T^{11}$$